Optical phonons in carbon nanotubes: Kohn anomalies, Peierls distortions, and dynamic effects

Abstract

We present a detailed study of the vibrational properties of single wall carbon nanotubes ͑SWNTs͒. The phonon dispersions of SWNTs are strongly shaped by the effects of electron-phonon coupling. We analyze the separate contributions of curvature and confinement. Confinement plays a major role in modifying SWNT phonons and is often more relevant than curvature. Due to their one-dimensional character, metallic tubes are expected to undergo Peierls distortions ͑PD͒ at T = 0 K. At finite temperature, PD are no longer present, but phonons with atomic displacements similar to those of the PD are affected by strong Kohn anomalies ͑KA͒. We investigate by density functional theory ͑DFT͒ KA and PD in metallic SWNTs with diameters up to 3 nm, in the electronic temperature range from 4 K to 3000 K. We then derive a set of simple formulas accounting for all the DFT results. Finally, we prove that the static approach, commonly used for the evaluation of phonon frequencies in solids, fails because of the SWNTs reduced dimensionality. The correct description of KA in metallic SWNTs can be obtained only by using a dynamical approach, beyond the adiabatic Born-Oppenheimer approximation, by taking into account nonadiabatic contributions. Dynamic effects induce significant changes in the occurrence and shape of Kohn anomalies. We show that the SWNT Raman G peak can only be interpreted considering the combined dynamic, curvature and confinement effects. We assign the G + and G − peaks of metallic SWNTs to TO ͑circumferential͒ and LO ͑axial͒ modes, respectively, the opposite of semiconducting SWNTs.

Non-adiabatic phonon dispersion of metallic single-walled carbon nanotubes

“…The calculated adiabatic phonon dispersion of the narrow metallic nanotube (8,5) with diameter of about 0.9 nm (Fig. 1) has Kohn anomalies in the form of smeared logarithmic softening of the LO(Γ) and TO(K) branches at the Brillouin zone center and at wave vector q * ≈ 0.21.…”

“…The nanotube curvature removes the degeneracy of the G band, splitting it into lower-and higher-frequency components usually denoted by G -and G + , respectively. The adiabatic LO(Γ) and TO(K) branches have Kohn anomalies in the form of logarithmic singularities at the Γ and K points [7,8]. The non-adiabatic LO(Γ) and TO(K) branches were found to have zero slopes at the Γ and K points and singularities close to these points [8].…”

“…These effects were found to be large for the longitudinal optical phonon, which gives rise to the G -band in the Raman spectra of metallic nanotubes, as well as for the carbon hexagon breathing phonon, which is of major importance for double resonance scattering processes in nanotubes. The theoretical work done so far focuses on the Kohn anomalies of the phonon dispersion close to these phonons using an electron zone-folding scheme within an ab initio approach and a chirality-independent curvature correction for the G --band frequencies [8]. From a practical point of view, knowledge of the precise diameter-and chirality-dependence of the Gband can be used to support the assignment of the Raman scattering data.…”

“…These processes effectively renormalize the phonons, a phenomenon known as the Kohn anomaly. In nanotubes, the renormalization is expressed in logarithmic softening and increased linewidth of certain phonon branches at low temperatures [8]. The correct description of this anomaly requires going beyond the adiabatic approximation and explicitly accounting for the dynamic effects.…”

“…For a long time, the intense high-frequency Raman G band has not been used for such purposes except for identification of the nanotubes as metallic or semiconducting [6]. The recent theoretical predictions of the G band frequency in the adiabatic approximation [7] and with dynamic corrections [8,9], have made it possible to use the experimental data from this band to support the nanotube characterization [10]. The behavior of the G band with varying temperature [11], strain [12], and charge doping [13] provides additional important information about the nanotubes under different conditions.…”